Q 51 - Calculus

Question

Find the work described in each of Exercises 49–60.

The work required to pump all of the water out of the top of a giant rectangular hot tub that is 5 feet wide, 8 feet long, and 3 feet deep.

Step-by-Step Solution

Answer

Required work is 11,232  foot-pounds.

Step 1: Given Information

The work required to pump all of the water out of the top of a giant rectangular hot tub that is 5 feet wide, 8 feet long, and 3 feet deep.

Step 2: Slice representation

Hot tub is  3 feet deep. Height of the tank is y=3  and bottom is y=0

Width of the slice is delta y or y_k

Slice at y needs to be removed  upward is 

$3-y_k$

Step 3: Volume and Work

Volume of tub= length times width times height 

Tub is 5 feet wide, 8 feet long  and change in width is delta y

$$\begin{gathered} V= 5*8*∆y \\ V=40∆y \\ Work required formula \\ W=Fd=\omega Vd \\ Water density = 62.4 pounds per foot \\ d=3-y_k \\ W=(62.4)(40∆y)\left(3-y_k\right) \end{gathered}$$

Step 4: Integration

Work required to pull out the water completely is the sum of the slices from y=0 to 3

$$\begin{gathered} W=(62.4)(40∆y)\left(3-y_k\right) \\ W={\int }_0^3(62.4)\left(40\right)\left(3-y\right)dy \\ =2496{\int }_0^3 \left(3-y\right)dy \\ =2496{\left[3y-\frac{y^2}{2}\right]}_0^3 \\ =2496\left[3\left(3\right)-\frac{3^2}{2}\right] \\ =11232 \end{gathered}$$